Well, the basic advice here is you watch movies and read books for inspiration, and take notes. The Robinson novels are full of good escort mission complications (not the island parts, of course). Lord of the Rings is full of good escort mission complications. Several centuries of adventure literature and decades of adventure movies are at your service.

A second very basic advice is you use pre-written adventure modules - either for inspiration only or as a basis for your adventures. Some of them are really good or at least have very good elements you can borrow.

I myself prefer moral complications in adventures. Say, the party has to escort the princess somewhere in a hurry but they stumble upon a village that needs to be saved from a monster. Will they stay and help the poor village people, or will they continue with their mission? Or they intercepted by the long-time rival of the prince, the evil Sir Lord Baron Von Vaderham and his retinue of clearly greater numbers. The Baron tells the party that if they hand him the prince, they can go freely and will also get a huge amount of treasure for their troubles. If they don't, they are surely to perish. They have to decide by next dawn. What do they do? This sort of complication is also a good step towards role-playing  instead of roll-playing.

Also, a trick I use, when I want to spice up things a bit, is to pretend something is up the players don't know about. You ask them their Perception scores and roll some dice behind the screen (or have them roll, doesn't count). Then you hmm a lot and pretend that you pretend that nothing is going on. And if they ask, you tell them its nothing, they don't see or hear a thing. Or you give them some little general detail, like "you hear a crack from among the trees, but that's nothing really". They sure are going to start to investigate and speculate, if not immediately, then after the second or third roll. Now the trick is this: you don't have to have anything up your sleeve, what is important is to listen closely to the players' discussion and their ideas of what this situation may be. Soon, they will have a very elaborated conspiracy theory of what is going on. Now this is an adventure hook the players give you, you can act upon it: go with it or directly against it, whatever best suits the situation.